--- a/src/HOL/IMP/Compiler.thy Thu Nov 22 23:46:33 2001 +0100
+++ b/src/HOL/IMP/Compiler.thy Fri Nov 23 17:19:14 2001 +0100
@@ -14,24 +14,24 @@
syntax
"@stepa1" :: "[instr list,state,nat,state,nat] => bool"
- ("_ |- <_,_>/ -1-> <_,_>" [50,0,0,0,0] 50)
+ ("_ \<turnstile> <_,_>/ -1\<rightarrow> <_,_>" [50,0,0,0,0] 50)
"@stepa" :: "[instr list,state,nat,state,nat] => bool"
- ("_ |-/ <_,_>/ -*-> <_,_>" [50,0,0,0,0] 50)
+ ("_ \<turnstile>/ <_,_>/ -*\<rightarrow> <_,_>" [50,0,0,0,0] 50)
-translations "P |- <s,m> -1-> <t,n>" == "((s,m),t,n) : stepa1 P"
- "P |- <s,m> -*-> <t,n>" == "((s,m),t,n) : ((stepa1 P)^*)"
+translations "P \<turnstile> <s,m> -1\<rightarrow> <t,n>" == "((s,m),t,n) : stepa1 P"
+ "P \<turnstile> <s,m> -*\<rightarrow> <t,n>" == "((s,m),t,n) : ((stepa1 P)^*)"
inductive "stepa1 P"
intros
-ASIN:
- "\<lbrakk> n<size P; P!n = ASIN x a \<rbrakk> \<Longrightarrow> P |- <s,n> -1-> <s[x::= a s],Suc n>"
-JMPFT:
- "\<lbrakk> n<size P; P!n = JMPF b i; b s \<rbrakk> \<Longrightarrow> P |- <s,n> -1-> <s,Suc n>"
-JMPFF:
- "\<lbrakk> n<size P; P!n = JMPF b i; ~b s; m=n+i \<rbrakk> \<Longrightarrow> P |- <s,n> -1-> <s,m>"
-JMPB:
- "\<lbrakk> n<size P; P!n = JMPB i; i <= n \<rbrakk> \<Longrightarrow> P |- <s,n> -1-> <s,n-i>"
+ASIN[simp]:
+ "\<lbrakk> n<size P; P!n = ASIN x a \<rbrakk> \<Longrightarrow> P \<turnstile> <s,n> -1\<rightarrow> <s[x::= a s],Suc n>"
+JMPFT[simp,intro]:
+ "\<lbrakk> n<size P; P!n = JMPF b i; b s \<rbrakk> \<Longrightarrow> P \<turnstile> <s,n> -1\<rightarrow> <s,Suc n>"
+JMPFF[simp,intro]:
+ "\<lbrakk> n<size P; P!n = JMPF b i; ~b s; m=n+i \<rbrakk> \<Longrightarrow> P \<turnstile> <s,n> -1\<rightarrow> <s,m>"
+JMPB[simp]:
+ "\<lbrakk> n<size P; P!n = JMPB i; i <= n; j = n-i \<rbrakk> \<Longrightarrow> P \<turnstile> <s,n> -1\<rightarrow> <s,j>"
consts compile :: "com => instr list"
primrec
@@ -50,96 +50,150 @@
of instructions; only needed for the first proof *)
lemma app_right_1:
- "is1 |- <s1,i1> -1-> <s2,i2> \<Longrightarrow> is1 @ is2 |- <s1,i1> -1-> <s2,i2>"
-apply(erule stepa1.induct);
- apply (simp add:ASIN)
- apply (force intro!:JMPFT)
- apply (force intro!:JMPFF)
-apply (simp add: JMPB)
-done
-
+ "is1 \<turnstile> <s1,i1> -1\<rightarrow> <s2,i2> \<Longrightarrow> is1 @ is2 \<turnstile> <s1,i1> -1\<rightarrow> <s2,i2>"
+ (is "?P \<Longrightarrow> _")
+proof -
+ assume ?P
+ then show ?thesis
+ by induct force+
+qed
+
lemma app_left_1:
- "is2 |- <s1,i1> -1-> <s2,i2> \<Longrightarrow>
- is1 @ is2 |- <s1,size is1+i1> -1-> <s2,size is1+i2>"
-apply(erule stepa1.induct);
- apply (simp add:ASIN)
- apply (fastsimp intro!:JMPFT)
- apply (fastsimp intro!:JMPFF)
-apply (simp add: JMPB)
-done
+ "is2 \<turnstile> <s1,i1> -1\<rightarrow> <s2,i2> \<Longrightarrow>
+ is1 @ is2 \<turnstile> <s1,size is1+i1> -1\<rightarrow> <s2,size is1+i2>"
+ (is "?P \<Longrightarrow> _")
+proof -
+ assume ?P
+ then show ?thesis
+ by induct force+
+qed
+
+declare rtrancl_induct2 [induct set: rtrancl]
lemma app_right:
- "is1 |- <s1,i1> -*-> <s2,i2> \<Longrightarrow> is1 @ is2 |- <s1,i1> -*-> <s2,i2>"
-apply(erule rtrancl_induct2);
- apply simp
-apply(blast intro:app_right_1 rtrancl_trans)
-done
+ "is1 \<turnstile> <s1,i1> -*\<rightarrow> <s2,i2> \<Longrightarrow> is1 @ is2 \<turnstile> <s1,i1> -*\<rightarrow> <s2,i2>"
+ (is "?P \<Longrightarrow> _")
+proof -
+ assume ?P
+ then show ?thesis
+ proof induct
+ show "is1 @ is2 \<turnstile> <s1,i1> -*\<rightarrow> <s1,i1>" by simp
+ next
+ fix s1' i1' s2 i2
+ assume "is1 @ is2 \<turnstile> <s1,i1> -*\<rightarrow> <s1',i1'>"
+ "is1 \<turnstile> <s1',i1'> -1\<rightarrow> <s2,i2>"
+ thus "is1 @ is2 \<turnstile> <s1,i1> -*\<rightarrow> <s2,i2>"
+ by(blast intro:app_right_1 rtrancl_trans)
+ qed
+qed
lemma app_left:
- "is2 |- <s1,i1> -*-> <s2,i2> \<Longrightarrow>
- is1 @ is2 |- <s1,size is1+i1> -*-> <s2,size is1+i2>"
-apply(erule rtrancl_induct2);
- apply simp
-apply(blast intro:app_left_1 rtrancl_trans)
-done
+ "is2 \<turnstile> <s1,i1> -*\<rightarrow> <s2,i2> \<Longrightarrow>
+ is1 @ is2 \<turnstile> <s1,size is1+i1> -*\<rightarrow> <s2,size is1+i2>"
+ (is "?P \<Longrightarrow> _")
+proof -
+ assume ?P
+ then show ?thesis
+ proof induct
+ show "is1 @ is2 \<turnstile> <s1,length is1 + i1> -*\<rightarrow> <s1,length is1 + i1>" by simp
+ next
+ fix s1' i1' s2 i2
+ assume "is1 @ is2 \<turnstile> <s1,length is1 + i1> -*\<rightarrow> <s1',length is1 + i1'>"
+ "is2 \<turnstile> <s1',i1'> -1\<rightarrow> <s2,i2>"
+ thus "is1 @ is2 \<turnstile> <s1,length is1 + i1> -*\<rightarrow> <s2,length is1 + i2>"
+ by(blast intro:app_left_1 rtrancl_trans)
+ qed
+qed
lemma app_left2:
- "\<lbrakk> is2 |- <s1,i1> -*-> <s2,i2>; j1 = size is1+i1; j2 = size is1+i2 \<rbrakk> \<Longrightarrow>
- is1 @ is2 |- <s1,j1> -*-> <s2,j2>"
+ "\<lbrakk> is2 \<turnstile> <s1,i1> -*\<rightarrow> <s2,i2>; j1 = size is1+i1; j2 = size is1+i2 \<rbrakk> \<Longrightarrow>
+ is1 @ is2 \<turnstile> <s1,j1> -*\<rightarrow> <s2,j2>"
by (simp add:app_left)
lemma app1_left:
- "is |- <s1,i1> -*-> <s2,i2> \<Longrightarrow>
- instr # is |- <s1,Suc i1> -*-> <s2,Suc i2>"
+ "is \<turnstile> <s1,i1> -*\<rightarrow> <s2,i2> \<Longrightarrow>
+ instr # is \<turnstile> <s1,Suc i1> -*\<rightarrow> <s2,Suc i2>"
by(erule app_left[of _ _ _ _ _ "[instr]",simplified])
-
+declare rtrancl_into_rtrancl[trans]
+ rtrancl_into_rtrancl2[trans]
+ rtrancl_trans[trans]
(* The first proof; statement very intuitive,
but application of induction hypothesis requires the above lifting lemmas
*)
-theorem "<c,s> -c-> t ==> compile c |- <s,0> -*-> <t,length(compile c)>"
-apply(erule evalc.induct);
- apply simp;
- apply(force intro!: ASIN);
- apply simp
- apply(rule rtrancl_trans)
- apply(erule app_right)
- apply(erule app_left[of _ 0,simplified])
-(* IF b THEN c0 ELSE c1; case b is true *)
- apply(simp);
- (* execute JMPF: *)
- apply (rule rtrancl_into_rtrancl2)
- apply(force intro!: JMPFT);
- (* execute compile c0: *)
- apply(rule app1_left)
- apply(rule rtrancl_into_rtrancl);
- apply(erule app_right)
- (* execute JMPF: *)
- apply(force intro!: JMPFF);
-(* end of case b is true *)
- apply simp
- apply (rule rtrancl_into_rtrancl2)
- apply(force intro!: JMPFF)
- apply(force intro!: app_left2 app1_left)
-(* WHILE False *)
- apply(force intro: JMPFF);
-(* WHILE True *)
-apply(simp)
-apply(rule rtrancl_into_rtrancl2);
- apply(force intro!: JMPFT);
-apply(rule rtrancl_trans);
- apply(rule app1_left)
- apply(erule app_right)
-apply(rule rtrancl_into_rtrancl2);
- apply(force intro!: JMPB)
-apply(simp)
-done
+theorem "<c,s> -c-> t \<Longrightarrow> compile c \<turnstile> <s,0> -*\<rightarrow> <t,length(compile c)>"
+ (is "?P \<Longrightarrow> ?Q c s t")
+proof -
+ assume ?P
+ then show ?thesis
+ proof induct
+ show "\<And>s. ?Q SKIP s s" by simp
+ next
+ show "\<And>a s x. ?Q (x :== a) s (s[x::= a s])" by force
+ next
+ fix c0 c1 s0 s1 s2
+ assume "?Q c0 s0 s1"
+ hence "compile c0 @ compile c1 \<turnstile> <s0,0> -*\<rightarrow> <s1,length(compile c0)>"
+ by(rule app_right)
+ moreover assume "?Q c1 s1 s2"
+ hence "compile c0 @ compile c1 \<turnstile> <s1,length(compile c0)> -*\<rightarrow>
+ <s2,length(compile c0)+length(compile c1)>"
+ proof -
+ note app_left[of _ 0]
+ thus
+ "\<And>is1 is2 s1 s2 i2.
+ is2 \<turnstile> <s1,0> -*\<rightarrow> <s2,i2> \<Longrightarrow>
+ is1 @ is2 \<turnstile> <s1,size is1> -*\<rightarrow> <s2,size is1+i2>"
+ by simp
+ qed
+ ultimately have "compile c0 @ compile c1 \<turnstile> <s0,0> -*\<rightarrow>
+ <s2,length(compile c0)+length(compile c1)>"
+ by (rule rtrancl_trans)
+ thus "?Q (c0; c1) s0 s2" by simp
+ next
+ fix b c0 c1 s0 s1
+ let ?comp = "compile(IF b THEN c0 ELSE c1)"
+ assume "b s0" and IH: "?Q c0 s0 s1"
+ hence "?comp \<turnstile> <s0,0> -1\<rightarrow> <s0,1>" by auto
+ also from IH
+ have "?comp \<turnstile> <s0,1> -*\<rightarrow> <s1,length(compile c0)+1>"
+ by(auto intro:app1_left app_right)
+ also have "?comp \<turnstile> <s1,length(compile c0)+1> -1\<rightarrow> <s1,length ?comp>"
+ by(auto)
+ finally show "?Q (IF b THEN c0 ELSE c1) s0 s1" .
+ next
+ fix b c0 c1 s0 s1
+ let ?comp = "compile(IF b THEN c0 ELSE c1)"
+ assume "\<not>b s0" and IH: "?Q c1 s0 s1"
+ hence "?comp \<turnstile> <s0,0> -1\<rightarrow> <s0,length(compile c0) + 2>" by auto
+ also from IH
+ have "?comp \<turnstile> <s0,length(compile c0)+2> -*\<rightarrow> <s1,length ?comp>"
+ by(force intro!:app_left2 app1_left)
+ finally show "?Q (IF b THEN c0 ELSE c1) s0 s1" .
+ next
+ fix b c and s::state
+ assume "\<not>b s"
+ thus "?Q (WHILE b DO c) s s" by force
+ next
+ fix b c and s0::state and s1 s2
+ let ?comp = "compile(WHILE b DO c)"
+ assume "b s0" and
+ IHc: "?Q c s0 s1" and IHw: "?Q (WHILE b DO c) s1 s2"
+ hence "?comp \<turnstile> <s0,0> -1\<rightarrow> <s0,1>" by auto
+ also from IHc
+ have "?comp \<turnstile> <s0,1> -*\<rightarrow> <s1,length(compile c)+1>"
+ by(auto intro:app1_left app_right)
+ also have "?comp \<turnstile> <s1,length(compile c)+1> -1\<rightarrow> <s1,0>" by simp
+ also note IHw
+ finally show "?Q (WHILE b DO c) s0 s2".
+ qed
+qed
(* Second proof; statement is generalized to cater for prefixes and suffixes;
needs none of the lifting lemmas, but instantiations of pre/suffix.
*)
theorem "<c,s> -c-> t ==>
- !a z. a@compile c@z |- <s,length a> -*-> <t,length a + length(compile c)>";
+ !a z. a@compile c@z \<turnstile> <s,length a> -*\<rightarrow> <t,length a + length(compile c)>";
apply(erule evalc.induct);
apply simp;
apply(force intro!: ASIN);